Gert de Cooman

POSSIBILITY THEORY I: THE MEASURE- AND INTEGRAL-THEORETIC GROUNDWORK

Abstract In this paper, I provide the basis for a measure- and integral-theoretic formulation of possibility theory. It is shown thai, using a general definition of possibility measures, and a generalization of Sugenos fuzzy integral-the semi-normed fuzzy integral, or possibility integral-. a unified and consistent account can be given of many of the possibilistic results extant in the literature. The striking formal analogy between this treatment of possibility theory, using possibility integrals, and Kolmogorovs measure-theoretic formulation of probability theory, using Lebesgue integrals, is explored and exploited. I introduce and study possibilistic and fuzzy variables as possibilistic counterparts of stochastic and real stochastic variables respeclively, and develop the notion of a possibility distribution for these variables. The almost everywhere equality and dominance of fuzzy variables is defined and studied. The proof is given for a Radon-Nikodym-like theorem in possibility theory. Following t...

Supremum preserving upper probabilities

Abstract We study the relation between possibility measures and the theory of imprecise probabilities, and argue that possibility measures have an important part in this theory. It is shown that a possibility measure is a coherent upper probability if and only if it is normal. A detailed comparison is given between the possibilistic and natural extension of an upper probability, both in the general case and for upper probabilities defined on a class of nested sets. We prove in particular that a possibility measure is the restriction to events of the natural extension of a special kind of upper probability, defined on a class of nested sets. We show that possibilistic extension can be interpreted in terms of natural extension. We also prove that when either the upper or the lower cumulative distribution function of a random quantity is specified, possibility measures very naturally emerge as the corresponding natural extensions. Next, we go from upper probabilities to upper previsions. We show that if a coherent upper prevision defined on the convex cone of all non-negative gambles is supremum preserving, then it must take the form of a Shilkret integral associated with a possibility measure. But at the same time, we show that such a supremum preserving upper prevision is never coherent unless it is the vacuous upper prevision with respect to a non-empty subset of the universe of discourse.

POSSIBILITY THEORY II: CONDITIONAL POSSIBILITY

Abstract It is shown that the nolion of conditional possibility can be consistently iniroduced in possibility theory, in very much the same way as conditional expectations and probabilities are defined in the measure- and integral-theoretic treatment of probability theory. I write down possibilistic integral equations which are formal counterparts of the integral equations used to define conditional expectations and probabilities, and use their solutions to define conditional possibilities. In all, three types of conditional possibilities, with special cases, are introduced and studied. I explain why, like conditional expectations, conditional possibilities are not uniquely defined, but can only be determined up to almost everywhere equality, and I assess the consequences of this nondeterminacy. I also show that this approach solves a number of consistency problems, extant in Ihe literature.

A behavioural model for vague probability assessments

I present an hierarchical uncertainty model that is able to represent vague probability assessments, and to make inferences based on them. This model can be given an interpretation in terms of the behaviour of a modeller in the face of uncertainty, and is based on Walleys theory of imprecise probabilities. It is formally closely related to Zadehs fuzzy probabilities, but it has a different interpretation, and a different calculus. Through rationality (coherence) arguments, the hierarchical model is shown to lead to an imprecise first-order uncertainty model that can be used in decision making, and as a prior in statistical reasoning.

POSSIBILITY THEORY III: POSSIBILISTIC INDEPENDENCE

Abstract The introduction of the notion of independence in possibility theory is a problem of long-standing interest. Many of the measure-theoretic definitions that have up to now been given in the literature face some difficulties as far as interpretation is concerned. Also, there are inconsistencies between the definition of independence of measurable sets and possibilistic variables. After a discussion of these definitions and their shortcomings, a new measure-theoretic definition is suggested, which is consistent in this respect, and which is a formal counterpart of the definition of stochastic independence in probability theory. In discussing the properties of possibilistic independence, I draw from the measure- and integral-theoretic treatment of possibility theory, discussed in Part I of this series of three papers. I also investigate the relationship between this definition of possibilistic independence and the definition of conditional possibility, discussed in detail in Part II of this series. F...

Imprecise markov chains and their limit behavior

When the initial and transition probabilities of a finite Markov chain in discrete time are not well known, we should perform a sensitivity analysis. This can be done by considering as basic uncertainty models the so-called credal sets that these probabilities are known or believed to belong to and by allowing the probabilities to vary over such sets. This leads to the definition of an imprecise Markov chain. We show that the time evolution of such a system can be studied very efficiently using so-called lower and upper expectations, which are equivalent mathematical representations of credal sets. We also study how the inferred credal set about the state at time n evolves as n→∞: under quite unrestrictive conditions, it converges to a uniquely invariant credal set, regardless of the credal set given for the initial state. This leads to a non-trivial generalization of the classical Perron–Frobenius theorem to imprecise Markov chains.

Epistemic irrelevance in credal nets: The case of imprecise Markov trees

We focus on credal nets, which are graphical models that generalise Bayesian nets to imprecise probability. We replace the notion of strong independence commonly used in credal nets with the weaker notion of epistemic irrelevance, which is arguably more suited for a behavioural theory of probability. Focusing on directed trees, we show how to combine the given local uncertainty models in the nodes of the graph into a global model, and we use this to construct and justify an exact message-passing algorithm that computes updated beliefs for a variable in the tree. The algorithm, which is linear in the number of nodes, is formulated entirely in terms of coherent lower previsions, and is shown to satisfy a number of rationality requirements. We supply examples of the algorithms operation, and report an application to on-line character recognition that illustrates the advantages of our approach for prediction. We comment on the perspectives, opened by the availability, for the first time, of a truly efficient algorithm based on epistemic irrelevance.

Imprecise probability trees: Bridging two theories of imprecise probability

We give an overview of two approaches to probability theory where lower and upper probabilities, rather than probabilities, are used: Walleys behavioural theory of imprecise probabilities, and Shafer and Vovks game-theoretic account of probability. We show that the two theories are more closely related than would be suspected at first sight, and we establish a correspondence between them that (i) has an interesting interpretation, and (ii) allows us to freely import results from one theory into the other. Our approach leads to an account of probability trees and random processes in the framework of Walleys theory. We indicate how our results can be used to reduce the computational complexity of dealing with imprecision in probability trees, and we prove an interesting and quite general version of the weak law of large numbers.

Marginal extension in the theory of coherent lower previsions

We generalise Walleys Marginal Extension Theorem to the case of any finite number of conditional lower previsions. Unlike the procedure of natural extension, our marginal extension always provides the smallest (most conservative) coherent extensions. We show that they can also be calculated as lower envelopes of marginal extensions of conditional linear (precise) previsions. Finally, we use our version of the theorem to study the so-called forward irrelevant product and forward irrelevant natural extension of a number of marginal lower previsions.

Precision–imprecision equivalence in a broad class of imprecise hierarchical uncertainty models

Abstract Hierarchical models are rather common in uncertainty theory. They arise when there is a ‘correct’ or ‘ideal’ (the so-called first-order ) uncertainty model about a phenomenon of interest, but the modeler is uncertain about what it is. The modelers uncertainty is then called second-order uncertainty . For most of the hierarchical models in the literature, both the first- and the second-order models are precise , i.e., they are based on classical probabilities. In the present paper, I propose a specific hierarchical model that is imprecise at the second level, which means that at this level, lower probabilities are used. No restrictions are imposed on the underlying first-order model: that is allowed to be either precise or imprecise. I argue that this type of hierarchical model generalizes and includes a number of existing uncertainty models, such as imprecise probabilities, Bayesian models, and fuzzy probabilities. The main result of the paper is what I call precision–imprecision equivalence : the implications of the model for decision making and statistical reasoning are the same, whether the underlying first-order model is assumed to be precise or imprecise.